Question: Simplify and expand the following expression: $ \dfrac{4x}{4x - 4}+\dfrac{10}{x - 7} $
Answer: In order to add expressions, they must have a common denominator. Get both fractions over a common denominator of $(4x - 4)(x - 7)$ Multiply the first term by $\dfrac{x - 7}{x - 7}$ $ \begin{align*} \dfrac{4x}{4x - 4} \times \dfrac{x - 7}{x - 7} & = \dfrac{(4x)(x - 7)}{(4x - 4)(x - 7)} \\ & = \dfrac{4x^2 - 28x}{(4x - 4)(x - 7)}\end{align*} $ Multiply the second term by $\dfrac{4x - 4}{4x - 4}$ $ \begin{align*} \dfrac{10}{x - 7} \times \dfrac{4x - 4}{4x - 4} & = \dfrac{(10)(4x - 4)}{(x - 7)(4x - 4)} \\ & = \dfrac{40x - 40}{(x - 7)(4x - 4)}\end{align*} $ Now we have: $ = \dfrac{4x^2 - 28x}{(4x - 4)(x - 7)} + \dfrac{40x - 40}{(x - 7)(4x - 4)} $ Now both terms have a common denominator we can simply add the numerators: $ = \dfrac{4x^2 - 28x + 40x - 40}{(4x - 4)(x - 7)} $ $ = \dfrac{4x^2 + 12x - 40}{(4x - 4)(x - 7)}$ Expand the denominator: $ = \dfrac{4x^2 + 12x - 40}{4x^2 - 32x + 28}$ Simplify: $ = \dfrac{x^2 + 3x - 10}{x^2 - 8x + 7}$